In Section REF  we analyze the first hypothesis how the sampling bias affects the PDP estimation. An indicator of the size of the sampling bias is the exploration factor \(\tau \) . The smaller \(\tau \)  the higher the sampling bias compared to a uniformly distributed dataset (e.g. see Figure REF ). To put it in other words, the sampling bias can be quantified by the distributional shift between a uniformly distributed sample and the sample generated by the BO process. A commonly used measure to quantify such distributional differences is the Kullback-Leibler divergence. However, since the joint distribution of the generated sample is not known, the Kullback-Leibler divergence might not be the most appropriate measure here.
Another metric that tests if two different samples belong to the same distribution, is the maximum mean discrepancy (MMD) [1]}. It is defined by the maximum deviation in expectation and based on the function class of reproducing kernel Hilbert space (RKHS). This is equivalent to `the norm of the difference between distribution feature means in the RKHS' [1]}.
